Question: Simplify the expression $x - 2(1+x) + 3(1-x) - 4(1+2x)$.
Since subtracting a term is the same as adding the negative, we have $x + [-2(1+x)] + 3(1-x) + [-4(1+2x)]$. Now, we can distribute several terms and negative signs.  We have $-2(1+x) = -2 -2x$, and $-4(1+2x) = -4 -8x$.  Also, $3(1-x) = 3 - 3x$.

Substituting for these simplified expressions, we get $x + (-2 -2x) + (3 - 3x) + (-4 -8x)$.  Next, we can group together similar terms by separating constants from the variable $x$.  So, we have $(x -2x -3x -8x) + (-2 +3 -4) = (-12x) + (-3)$.  This yields $\boxed{-12x -3}$.